Question 1172783: A truck manufacturer monitors the width of the door seam as vehicles come off its assembly line. The seam width is the distance between the edge of the door and the truck body, in inches. These data are 62 days of measurements of a passenger door seam, with 10 trucks measured each day. It has been claimed that the process has an average width 0.275 with sigma.png = 0.1.
(a) If the seam widths at this assembly line are normally distributed, then what is the probability of finding a seam wider than 1/2 inch?
(b) If the process is under control, what is the probability of finding the mean of a daily sample of 10 widths more than 3 standard errors away from mu.png= 0.275?
(c) Ten measurements are averaged each day. Is this a large enough sample size to justify using a normal model to set the limits in the X-bar chart? Do you recommend changes in future testing?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**Given:**
* Claimed mean (μ) = 0.275 inches
* Standard deviation (σ) = 0.1 inches
* Sample size per day (n) = 10 trucks
* Number of days = 62
**a. Probability of a Seam Wider Than 0.5 Inches**
1. **Calculate the Z-score:**
* Z = (X - μ) / σ
* X = 0.5 inches
* Z = (0.5 - 0.275) / 0.1
* Z = 0.225 / 0.1
* Z = 2.25
2. **Find the probability:**
* We want to find P(X > 0.5) or P(Z > 2.25).
* Using a standard normal distribution table or calculator, we find that P(Z ≤ 2.25) ≈ 0.9878.
* Therefore, P(Z > 2.25) = 1 - 0.9878 = 0.0122.
* The probability of finding a seam wider than 0.5 inches is approximately 0.0122.
**b. Probability of a Daily Sample Mean More Than 3 Standard Errors Away From μ = 0.275**
1. **Standard error of the mean (SEM):**
* SEM = σ / √n
* SEM = 0.1 / √10
* SEM ≈ 0.0316
2. **3 Standard errors:**
* 3 * SEM = 3 * 0.0316 ≈ 0.0948
3. **Find the limits:**
* Upper limit: μ + 3 * SEM = 0.275 + 0.0948 ≈ 0.3698
* Lower limit: μ - 3 * SEM = 0.275 - 0.0948 ≈ 0.1802
4. **Probability:**
* We want to find P(X̄ < 0.1802) + P(X̄ > 0.3698).
* Since the daily sample mean (X̄) is normally distributed (Central Limit Theorem applies), we can use the standard normal distribution.
* Z-score for the upper limit: (0.3698 - 0.275) / 0.0316 ≈ 3
* Z-score for the lower limit: (0.1802 - 0.275) / 0.0316 ≈ -3
* P(Z < -3) + P(Z > 3) = 0.0013 + 0.0013 = 0.0026
* Therefore, the probability of finding the mean of a daily sample of 10 widths more than 3 standard errors away from 0.275 is approximately 0.0026.
**c. Sample Size and X-bar Chart**
1. **Justification of Normal Model:**
* The Central Limit Theorem (CLT) states that for a sufficiently large sample size, the distribution of the sample mean will be approximately normal, regardless of the population distribution.
* While n = 10 is not considered a large sample by some standards, if the population distribution of seam widths is reasonably close to normal (as stated in part a), then the distribution of the sample mean will also be reasonably close to normal.
* Given that the population is stated to be normally distributed, the sample size of 10 is sufficient.
2. **Recommendations for Future Testing:**
* **Increase Sample Size:** While n = 10 is acceptable, increasing the sample size per day would provide a more accurate representation of the process and improve the sensitivity of the X-bar chart. A sample size of 30 or more is generally considered ideal.
* **Monitor Process Stability:** In addition to the X-bar chart, consider using an R-chart (range chart) to monitor the variability of the process. This will help detect changes in the process standard deviation.
* **Regular Review:** Periodically review the control limits of the X-bar chart to ensure they are still appropriate. If the process changes, the control limits may need to be recalculated.
* **Data Collection Automation:** If possible, automate the data collection process to reduce the risk of human error and increase the efficiency of data collection.
**Summary:**
* **(a)** Approximately 0.0122
* **(b)** Approximately 0.0026
* **(c)** The sample size of 10 is sufficient due to the population being normally distributed. Increasing the sample size, monitoring process stability, regular reviews, and automated data collection are recommended.
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